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Properties

Label 135.4.a.f.1.1
Level $135$
Weight $4$
Character 135.1
Self dual yes
Analytic conductor $7.965$
Analytic rank $0$
Dimension $3$
CM no
Inner twists $1$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [135,4,Mod(1,135)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("135.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(135, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 135 = 3^{3} \cdot 5 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 135.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [3,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.96525785077\)
Analytic rank: \(0\)
Dimension: \(3\)
Coefficient field: 3.3.5637.1
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{3} - x^{2} - 23x + 6 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 3 \)
Twist minimal: yes
Fricke sign: \(+1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(5.20067\) of defining polynomial
Character \(\chi\) \(=\) 135.1

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q-5.20067 q^{2} +19.0470 q^{4} +5.00000 q^{5} +24.4013 q^{7} -57.4517 q^{8} -26.0034 q^{10} +28.9839 q^{11} -65.3919 q^{13} -126.903 q^{14} +146.411 q^{16} -68.1718 q^{17} +104.424 q^{19} +95.2349 q^{20} -150.736 q^{22} +154.807 q^{23} +25.0000 q^{25} +340.082 q^{26} +464.772 q^{28} +205.658 q^{29} -18.2497 q^{31} -301.824 q^{32} +354.539 q^{34} +122.007 q^{35} -337.613 q^{37} -543.076 q^{38} -287.258 q^{40} +195.969 q^{41} +334.882 q^{43} +552.055 q^{44} -805.098 q^{46} +5.00398 q^{47} +252.425 q^{49} -130.017 q^{50} -1245.52 q^{52} +319.965 q^{53} +144.919 q^{55} -1401.90 q^{56} -1069.56 q^{58} -430.611 q^{59} +594.581 q^{61} +94.9106 q^{62} +398.396 q^{64} -326.960 q^{65} +195.876 q^{67} -1298.47 q^{68} -634.517 q^{70} -425.955 q^{71} +929.193 q^{73} +1755.82 q^{74} +1988.96 q^{76} +707.245 q^{77} +24.4296 q^{79} +732.057 q^{80} -1019.17 q^{82} +545.859 q^{83} -340.859 q^{85} -1741.61 q^{86} -1665.17 q^{88} +84.1332 q^{89} -1595.65 q^{91} +2948.60 q^{92} -26.0241 q^{94} +522.121 q^{95} +827.613 q^{97} -1312.78 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3 q - q^{2} + 23 q^{4} + 15 q^{5} + 44 q^{7} - 36 q^{8} - 5 q^{10} + 38 q^{11} + 28 q^{13} - 108 q^{14} + 191 q^{16} - 19 q^{17} + 187 q^{19} + 115 q^{20} + 122 q^{22} - 81 q^{23} + 75 q^{25} + 416 q^{26}+ \cdots - 2693 q^{98}+O(q^{100}) \) Copy content Toggle raw display

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\). You can download additional coefficients here.



Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −5.20067 −1.83871 −0.919357 − 0.393424i \(-0.871291\pi\)
−0.919357 + 0.393424i \(0.871291\pi\)
\(3\) 0 0
\(4\) 19.0470 2.38087
\(5\) 5.00000 0.447214
\(6\) 0 0
\(7\) 24.4013 1.31755 0.658774 − 0.752341i \(-0.271075\pi\)
0.658774 + 0.752341i \(0.271075\pi\)
\(8\) −57.4517 −2.53903
\(9\) 0 0
\(10\) −26.0034 −0.822298
\(11\) 28.9839 0.794451 0.397226 − 0.917721i \(-0.369973\pi\)
0.397226 + 0.917721i \(0.369973\pi\)
\(12\) 0 0
\(13\) −65.3919 −1.39511 −0.697556 − 0.716530i \(-0.745729\pi\)
−0.697556 + 0.716530i \(0.745729\pi\)
\(14\) −126.903 −2.42260
\(15\) 0 0
\(16\) 146.411 2.28768
\(17\) −68.1718 −0.972593 −0.486296 − 0.873794i \(-0.661653\pi\)
−0.486296 + 0.873794i \(0.661653\pi\)
\(18\) 0 0
\(19\) 104.424 1.26087 0.630435 − 0.776242i \(-0.282876\pi\)
0.630435 + 0.776242i \(0.282876\pi\)
\(20\) 95.2349 1.06476
\(21\) 0 0
\(22\) −150.736 −1.46077
\(23\) 154.807 1.40345 0.701727 − 0.712446i \(-0.252413\pi\)
0.701727 + 0.712446i \(0.252413\pi\)
\(24\) 0 0
\(25\) 25.0000 0.200000
\(26\) 340.082 2.56521
\(27\) 0 0
\(28\) 464.772 3.13691
\(29\) 205.658 1.31689 0.658443 − 0.752631i \(-0.271215\pi\)
0.658443 + 0.752631i \(0.271215\pi\)
\(30\) 0 0
\(31\) −18.2497 −0.105734 −0.0528668 − 0.998602i \(-0.516836\pi\)
−0.0528668 + 0.998602i \(0.516836\pi\)
\(32\) −301.824 −1.66736
\(33\) 0 0
\(34\) 354.539 1.78832
\(35\) 122.007 0.589226
\(36\) 0 0
\(37\) −337.613 −1.50009 −0.750044 − 0.661387i \(-0.769968\pi\)
−0.750044 + 0.661387i \(0.769968\pi\)
\(38\) −543.076 −2.31838
\(39\) 0 0
\(40\) −287.258 −1.13549
\(41\) 195.969 0.746469 0.373234 − 0.927737i \(-0.378249\pi\)
0.373234 + 0.927737i \(0.378249\pi\)
\(42\) 0 0
\(43\) 334.882 1.18765 0.593826 − 0.804594i \(-0.297617\pi\)
0.593826 + 0.804594i \(0.297617\pi\)
\(44\) 552.055 1.89149
\(45\) 0 0
\(46\) −805.098 −2.58055
\(47\) 5.00398 0.0155299 0.00776496 − 0.999970i \(-0.497528\pi\)
0.00776496 + 0.999970i \(0.497528\pi\)
\(48\) 0 0
\(49\) 252.425 0.735934
\(50\) −130.017 −0.367743
\(51\) 0 0
\(52\) −1245.52 −3.32158
\(53\) 319.965 0.829256 0.414628 − 0.909991i \(-0.363912\pi\)
0.414628 + 0.909991i \(0.363912\pi\)
\(54\) 0 0
\(55\) 144.919 0.355289
\(56\) −1401.90 −3.34529
\(57\) 0 0
\(58\) −1069.56 −2.42138
\(59\) −430.611 −0.950182 −0.475091 − 0.879937i \(-0.657585\pi\)
−0.475091 + 0.879937i \(0.657585\pi\)
\(60\) 0 0
\(61\) 594.581 1.24800 0.624002 − 0.781422i \(-0.285505\pi\)
0.624002 + 0.781422i \(0.285505\pi\)
\(62\) 94.9106 0.194414
\(63\) 0 0
\(64\) 398.396 0.778118
\(65\) −326.960 −0.623913
\(66\) 0 0
\(67\) 195.876 0.357166 0.178583 − 0.983925i \(-0.442849\pi\)
0.178583 + 0.983925i \(0.442849\pi\)
\(68\) −1298.47 −2.31562
\(69\) 0 0
\(70\) −634.517 −1.08342
\(71\) −425.955 −0.711994 −0.355997 − 0.934487i \(-0.615859\pi\)
−0.355997 + 0.934487i \(0.615859\pi\)
\(72\) 0 0
\(73\) 929.193 1.48978 0.744889 − 0.667188i \(-0.232502\pi\)
0.744889 + 0.667188i \(0.232502\pi\)
\(74\) 1755.82 2.75824
\(75\) 0 0
\(76\) 1988.96 3.00197
\(77\) 707.245 1.04673
\(78\) 0 0
\(79\) 24.4296 0.0347917 0.0173959 − 0.999849i \(-0.494462\pi\)
0.0173959 + 0.999849i \(0.494462\pi\)
\(80\) 732.057 1.02308
\(81\) 0 0
\(82\) −1019.17 −1.37254
\(83\) 545.859 0.721877 0.360938 − 0.932590i \(-0.382456\pi\)
0.360938 + 0.932590i \(0.382456\pi\)
\(84\) 0 0
\(85\) −340.859 −0.434957
\(86\) −1741.61 −2.18375
\(87\) 0 0
\(88\) −1665.17 −2.01714
\(89\) 84.1332 0.100203 0.0501017 − 0.998744i \(-0.484045\pi\)
0.0501017 + 0.998744i \(0.484045\pi\)
\(90\) 0 0
\(91\) −1595.65 −1.83813
\(92\) 2948.60 3.34144
\(93\) 0 0
\(94\) −26.0241 −0.0285551
\(95\) 522.121 0.563879
\(96\) 0 0
\(97\) 827.613 0.866303 0.433152 − 0.901321i \(-0.357401\pi\)
0.433152 + 0.901321i \(0.357401\pi\)
\(98\) −1312.78 −1.35317
\(99\) 0 0
Currently showing only \(a_p\); display all \(a_n\) Currently showing all \(a_n\); display only \(a_p\)

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 135.4.a.f.1.1 ✓ 3
3.2 odd 2 135.4.a.g.1.3 yes 3
4.3 odd 2 2160.4.a.bm.1.1 3
5.2 odd 4 675.4.b.l.649.1 6
5.3 odd 4 675.4.b.l.649.6 6
5.4 even 2 675.4.a.r.1.3 3
9.2 odd 6 405.4.e.r.271.1 6
9.4 even 3 405.4.e.t.136.3 6
9.5 odd 6 405.4.e.r.136.1 6
9.7 even 3 405.4.e.t.271.3 6
12.11 even 2 2160.4.a.be.1.1 3
15.2 even 4 675.4.b.k.649.6 6
15.8 even 4 675.4.b.k.649.1 6
15.14 odd 2 675.4.a.q.1.1 3
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
135.4.a.f.1.1 ✓ 3 1.1 even 1 trivial
135.4.a.g.1.3 yes 3 3.2 odd 2
405.4.e.r.136.1 6 9.5 odd 6
405.4.e.r.271.1 6 9.2 odd 6
405.4.e.t.136.3 6 9.4 even 3
405.4.e.t.271.3 6 9.7 even 3
675.4.a.q.1.1 3 15.14 odd 2
675.4.a.r.1.3 3 5.4 even 2
675.4.b.k.649.1 6 15.8 even 4
675.4.b.k.649.6 6 15.2 even 4
675.4.b.l.649.1 6 5.2 odd 4
675.4.b.l.649.6 6 5.3 odd 4
2160.4.a.be.1.1 3 12.11 even 2
2160.4.a.bm.1.1 3 4.3 odd 2